The positive solutions for a class of Kirchhoff‐type problems with critical Sobolev exponents on a bounded domain

We study the positive solutions for a class of Kirchhoff‐type problems involving the nonlinearity on a bounded domain. The major difficulty of such problems is the nonlinearity does not satisfy the Ambrosetti–Rabinowitz condition; especially, we cannot use Pohozaev's identity directly since our domain is bounded and the weight potentials are not ‐smoothness. Another difficulty is caused by the absence of compactness as the appearance of the critical Sobolev growth. In this work, we shall combine the Nehari manifold and some novel analytical skills to overcome the above difficulties and then obtain some existence results. Furthermore, we show some asymptotic behaviors of the solutions.